Best constants in Kahane-Khintchine inequalities for complex Steinhaus functions

Abstract

Let { φ k } k ≥ 1 {\{ {\varphi _k}\} _{k \geq 1}} be a sequence of independent random variables uniformly distributed on [ 0 , 2 π [ [0,2\pi [ , and let ‖ ∙ ‖ ψ {\left \| \bullet \right \|_\psi } denote the Orlicz norm induced by the function ψ ( x ) = exp ⁡ ( | x | 2 ) − 1 \psi (x) = \exp (|x{|^2}) - 1 . Then \[ ‖ ∑ k = 1 n z k e i φ k ‖ ψ ≤ 2 ( ∑ k = 1 n | z k | 2 ) 1 / 2 {\left \| {\sum \limits _{k = 1}^n {{z_k}{e^{i{\varphi _k}}}} } \right \|_\psi } \leq \sqrt 2 {\left ( {\sum \limits _{k = 1}^n {|{z_k}{|^2}} } \right )^{1/2}} \] for all z 1 , … , z n ∈ C {z_1}, \ldots ,{z_n} \in {\mathbf {C}} and all n ≥ 1 n \geq 1 . The constant 2 \sqrt 2 is shown to be the best possible. The method of proof relies upon a combinatorial argument, Taylor expansion, and the central limit theorem. The result is additionally strengthened by showing that the underlying functions are Schur-concave. The proof of this fact uses a result on the multinomial distribution of Rinott, and Schur’s proposition on the sum of convex functions. The estimates obtained throughout are shown to be the best possible. The result extends and generalizes to provide similar inequalities and estimates for other Orlicz norms.